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Suppose that $X=\{X_t, t\ge 0\}$ is a supercritical superprocess on a locally compact separable metric space $(E, m)$. Suppose that the spatial motion of $X$ is a Hunt process satisfying certain conditions and that the branching mechanism is of the form $$ \psi(x,\lambda)=-a(x)\lambda+b(x)\lambda^2+\int_{(0,+\infty)}(e^{-\lambda y}-1+\lambda y)n(x,dy), \quad x\in E, \quad\lambda> 0, $$ where $a\in \mathcal{B}_b(E)$, $b\in \mathcal{B}_b^+(E)$ and $n$ is a kernel from $E$ to $(0,\infty)$ satisfying $$ \sup_{x\in E}\int_0^\infty y^2 n(x,dy)<\infty. $$ Put $T_tf(x)=\mathbb{P}_{\delta_x}< f,X_t>$. Let $\lambda_0>0$ be the largest eigenvalue of the generator $L$ of $T_t$, and $\phi_0$ and $\hat{\phi}_0$ be the eigenfunctions of $L$ and $\hat{L}$ (the dural of $L$) respectively associated with $\lambda_0$. Under some conditions on the spatial motion and the $\phi_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $$ \lim_{t\rightarrow\infty}e^{-\lambda_0 t}< f, X_t> = W_\infty\int_E\hat{\phi}_0(y)f(y)m(dy),\quad \mathbb{P}_{\mu}{-a.s.}, $$ for any finite initial measure $\mu$ on $E$ with compact support, where $W_\infty$ is the martingale limit defined by $W_\infty:=\lim_{t\to\infty}e^{-\lambda_0t}< \phi_0, X_t>$. Moreover, the exceptional set in the above limit does not depend on the initial measure $\mu$ and the function $f$.

Suppose $d\ge 2$ and $0<\beta<\alpha<2$. We consider the non-local operator $\mathcal{L}^{b}=\Delta^{\alpha/2}+\mathcal{S}^{b}$, where $$\mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right)\frac{b(x,z)}{|z|^{d+\beta}}\,dy.$$ Here $b(x,z)$ is a bounded measurable function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that is symmetric in $z$, and $\mathcal{A}(d,-\beta)$ is a normalizing constant so that when $b(x, z)\equiv 1$, $\mathcal{S}^{b}$ becomes the fractional Laplacian $\Delta^{\beta/2}:=-(-\Delta)^{\beta/2}$. In other words, $$\mathcal{L}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right) j^b(x, z)\,dz,$$ where $j^b(x, z):= \mathcal{A}(d,-\alpha) |z|^{-(d+\alpha)}+ \mathcal{A}(d,-\beta) b(x, z)|z|^{-(d+\beta)}$. It is recently established in Chen and Wang [arXiv:1312.7594 [math.PR]] that, when $j^b(x, z)\geq 0$ on $\mathbb{R}^d\times \mathbb{R}^d$, there is a conservative Feller process $X^{b}$ having $\mathcal{L}^b$ as its infinitesimal generator. In this paper we establish, under certain conditions on $b$, a uniform boundary Harnack principle for harmonic functions of $X^b$ (or equivalently, of $\mathcal{L}^b$) in any $\kappa$-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of $X^{b}$ in open sets.

In this paper, we establish some functional central limit theorems for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. In the particular case when the state $E$ is a finite set and the underline motion is an irreducible Markov chain on $E$, our results are superprocess analogs of the functional central limit theorems of \cite{Janson} for supercritical multitype branching processes. The results of this paper are refinements of the central limit theorems in \cite{RSZ3}., Comment: arXiv admin note: text overlap with arXiv:1310.5410

In this paper, we establish a spatial central limit theorem for a large class of supercritical branching, not necessarily symmetric, Markov processes with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in \cite{RSZ2} for supercritical branching symmetric Markov processes. To prove our central limit theorem, we have to carefully develop the spectral theory of nonsymmetric strongly continuous semigroups which should be of independent interest.

In this paper we establish some conditional limit theorems for some critical superprocesses $X=\{X_t, t\ge 0\}$. First we identify the rate of non-extinction. Then we show that, for a large class of functions $f$, conditioned on non-extinction at time $t$, the limit, as $t\to\infty$, of $t^{-1}\langle f, X_t\rangle$ exists in distribution and we identify this limit. Finally, we also establish, under some conditions, a central limit theorem for $\langle f, X_t\rangle$ conditioned on non-extinction at time $t$.

In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in Mi{\l}o\'{s} (2012, arXiv:1203:6661) and Ren, Song and Zhang (2013, to appear in Acta Appl. Math., DOI 10.1007/s10440-013-9837-0) for supercritical super Ornstein-Uhlenbeck processes. The advantage of this central limit theorem is that it allows us to characterize the limit Gaussian field. In the case of supercritical super Ornstein-Uhlenbeck processes with non-spatially dependent branching mechanisms, our central limit theorem reveals more independent structures of the limit Gaussian field.

In this paper we study the conditional limit theorems for critical continuous-state branching processes with branching mechanism $\psi(\lambda)=\lambda^{1+\alpha}L(1/\lambda)$ where $\alpha\in [0,1]$ and $L$ is slowly varying at $\infty$. We prove that if $\alpha\in (0,1]$, there are norming constants $Q_{t}\to 0$ (as $t\uparrow +\infty$) such that for every $x>0$, $P_{x}\left(Q_{t}X_{t}\in\cdot|X_{t}>0\right)$ converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of $\psi$ at 0. We give a conditional limit theorem for the case $\alpha=0$. The limit theorems we obtain in this paper allow infinite variance of the branching process.

In this paper we study supercritical super-OU processes with general branching mechanisms satisfying a second moment condition. We establish central limit theorems for the super-OU processes. In the small and crtical branching rate cases, our central limit theorems sharpen the corresponding results in the recent preprint of Milos in that the limit normal random variables in our central limit theorems are non-degenerate. Our central limit theorems in the large branching rate case are completely new. The main tool of the paper is the so called "backbone decomposition" of superprocesses.

Consider a superdiffusion $X$ on $\mathbb R^d$ corresponding to the semilinear operator $\mathcal{A}(u)=Lu+\beta u-ku^2,$ where $L$ is a second order elliptic operator, $\beta(\cdot)$ is in the Kato class and bounded from above, and $k(\cdot)\ge 0$ is bounded on compact subsets of $\R^d$ and is positive on a set of positive Lebesgue measure. The main purpose of this paper is to complement the results obtained in \cite{Englander:2004}, in the following sense. Let $\lambda_\infty $ be the $L^\infty$-growth bound of the semigroup corresponding to the Schr\"odinger operator $L+\beta $. If $\lambda_\infty \neq0$, then we prove that, in some sense, the exponential growth/decay rate of $\|X_t\|$, the total mass of $X_t$, is $\lambda_\infty $. We also describe the limiting behavior of $\exp(-\lambda_\infty t)\|X_t\|$ in these cases. This should be compared to the result in \cite{Englander:2004}, which says that the generalized principal eigenvalue $\lambda_2$ of the operator gives the rate of {\it local} growth when it is positive, and implies local extinction otherwise. It is easy to show that $\lambda_{\infty}\ge \lambda_2$, and we discuss cases when $\lambda_{\infty}> \lambda_2$ and when $\lambda_{\infty}= \lambda_2$. When $\lambda_\infty =0$, and under some conditions on $\beta$, we give a sufficient and necessary condition for the superdiffusion $X$ to exhibit weak extinction. We show that the branching intensity $k$ affects weak extinction; this should be compared to the known result that $k$ does not affect weak {\it local} extinction (which only depends on the sign of $\lambda_2$, and which turns out to be equivalent to local extinction) of $X$.

We consider a supercritical Galton-Watson branching process with immigration. It is well known that under suitable conditions on the offspring and immigration distributions, there is a finite, strictly positive limit ${\mathcal{W}}$ for the normalized population size. Small value probabilities for ${\mathcal{W}}$ are obtained. Precise effects of the balance between offspring and immigration distributions are characterized., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ490 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

In this paper we prove that, under certain conditions, a strong law of large numbers holds for a class of super-diffusions $X$ corresponding to the evolution equation $\partial_t u_t=L u_t+\beta u_t-\psi(u_t)$ on a bounded domain $D$ in $\R^d$, where $L$ is the generator of the underlying diffusion and the branching mechanism $\psi(x,\lambda)=1/2\alpha(x)\lambda^2+\int_0^\infty (e^{-\lambda r}-1+\lambda r)n(x, {\rm d}r)$ satisfies $\sup_{x\in D}\int_0^\infty (r\wedge r^2) n(x,{\rm d}r)<\infty$.

In this paper we use the spine decomposition and martingale change of measure to establish a Kesten-Stigum $L\log L$ theorem for branching Hunt processes. This result is a generalization of the results in Asmussen-Hering (1976) and Hering (1978) for branching diffusions.